Digital Darkfield Decompositions

This is working draft of a note on
prospects for digital implementation of optical
darkfield strategies. For decades analog (electron optical) versions of
these have played key roles in the microscopy of materials.
Applications here are described in the context of recent
developments in mathematical harmonic analysis. Examples of
work with electron phase contrast images of heterostructure on the nanoscale
are offered. These push present day limits by simply using
sharp-edged Fourier windows with their foibles intact.
Comments, corrections, and contributions both on this communication,
and on ways to optimize the performance of these and related tools for future
application, are invited.

The "two-crystal" image on the
left may be helpful in learning to use digital darkfield routines to
map the location of periodicities in an image. You might begin
by comparing the power spectrum of that image (below left) with the
corresponding digital darkfield tableau (the tiled array of 16×16-1=255
log complex color darkfield images below right) to help figure out
what part of the image gives rise to each feature. Note for example
that some regions in the power spectrum "light up" the lower
elongate crystal, while others only light up the top crystal. The technique
may also work on "Where's Waldo" puzzles. If Waldo's shirt shows its
predictable pattern of horizontal stripes, see what lights up when you
use a downward pointing g-vector with the shirt's spatial frequency.
Challenges of periodicity location typically utilize
unexpected or faint patterns in the digital darkfield tableau, or
amplitude variations across specific darkfield images. Applications in
microscopy have included discovery of weak periodicities (like a
hidden iscosahedral twin or cross-fringe crucial to
phase identification) and quantitative mapping of
periodicity strengths (e.g. interface transitions or nanotube
walls in cross-section) from point to point in an image.

The "coherent inclusion" image at top right, on the other hand,
might be helpful in learning to quantitatively map strains.
You could also start with
an image of your favorite brick wall, as these tools
can rapidly pinpoint places where the mortar is starting
to yield. Gradients in the phase portion of a periodicity's
complex darkfield image make possible a
kind of digital interferometry. By way of applications in
microscopy, the above inclusion's
classic "line of no contrast" perpendicular to a lattice
periodicity (in transmission electron microscope darkfield images,
and brightfield images with one
active reflection) arises from the bimodal distribution of
strain in the g-vector direction as shown below.
Other microscopy applications include strain relaxation
in layered heterostructures (e.g.
sSi/Ge for
next generation computer chips), as well as
lattice parameter changes near: (i) defects in and
(ii) surfaces of nanoparticles. On extended lattices these
mapping techniques are routinely sensitive to displacements
hundreds of times smaller than image resolution.
In all cases, the concept of "active g-vector" or reference
frequency can be crucial to understanding what the data has to say.

The strain mapping macro requires that many of
the plugins above be compiled and present in your ImageJ plugins
directory, but it needs only input on aperture-size and a mouseclick
(to specify the reference periodicity) before generating
four standalone images and four x-y image pairs. These are namely: the
(i) power spectrum, (ii) darkfield amplitude, (iii)
darkfield phase, (iv) composite log color darkfield; along with: the
(v) complex Fourier transform, (vi) complex darkfield, (vii) xy gradient and
(viii) isotropic-shear strain maps. Screen caps generated with selected
g-vectors from
both model images above are shown below, although of course the routines
are designed for use with experimental images as well. The strain maps use
"orthogonal coordinate" colors, i.e. red-cyan for isotropic
compression-tension
parallel to the reference periodicity, and indigo-chartreuse for
counterclockwise-clockwise
shear perpendicular to it. Electron
microscopists often superpose the g-vector on these images to aid
interpretation.

Two experimental images that may be fun to play with are
here
and
here.
Look for extended versions of these and other ImageJ
routines here soon./pf2007july17

Optical darkfield imaging in
microscopy involves forming images of a specimen
(below left) using a back focal-plane
(scattering angle) aperture that excludes the unscattered beam.
It's called "dark field" because the field surrounding the
specimen doesn't scatter, so it's dark. The digital
darkfield animation below illustrates this by placing an aperture
(centered in the orange figure below) over the power spectrum
(a digital substitute for the back focal-plane's optical
diffraction pattern) here
shown with the DC peak (or unscattered beam) below center.
In this example, only nanocystals with
projected periodicities that diffract into the aperture
light up in the darkfield image at right, and this varies
with aperture position. In this case, the aperture is
moving by 1.25 degree increments around the ring associated
with diffraction from gold 2.3 Ångstrom (111) lattice
spacings. A larger traverse at 2.5 degree increments can
be found here.

These led e.g. to the lattice fringe (and diffracted electron) intensity maps below.

Here's a snapshot from application of digital darkfield periodicity
mapping techniques to columnar
quasi-epitaxial growth of Cu2O on single crystal silicon, synthesized in
Jay Switzer's lab at UM-R.

Here's a snapshot from the application of digital darkfield periodicity
and strain mapping to hidden
icosahedral twins,
which we've for example seen in electro-active polymers developed by
a company in the St. Louis area as well as in metal nanoparticles prepared
by Max Bertino's group at UM-R.

* In addition to outlining a quantitation process for
considering multiple reflections from the same image, Martin's maps
of three at-first-glance-invisible quantum dots in a HREM image
(below left) detail a 9.0 picometer (0.09Å) increase in GaInP's
d002 = 282.6 picometer (2.8Å)
lattice spacing as one moves across alternating
Ga0.52In0.48P and InP layers only ~2 nm
(20Å) in width (below right). Fringe spacings found on the InP
bands are typically within 1.9 picometers (0.019Å) of
the expected values for pure InP relative to matrix, if we
ignore averaging effects likely to bring theory even more closely
into agreement with Martin's measurements. The
contours show that this amounts to a local increase in
fringe spacing from about 10.98 to 11.33 image
pixels (each spanning ~25.0 picometers or ¼ Å), as one
moves from the GaInP matrix into an InP quantum dot. This is precise work
for digital interferometry on
images whose point resolution is near 200 picometers (2Å), and
only possible on images recorded in parallel (rather than with STEM) in
the face of picoscale drift.

This page is
http://www.umsl.edu/~fraundor/dgtldfld.html. Although there are many
contributors, the person responsible for errors is
P. Fraundorf. This
site is hosted by the Department of Physics and Astronomy (and Center for
NanoScience) at UM-StL.
Mindquilts
site page requests est. around 2000/day,
hence more than 500,000/year. Requests for
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